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• Function-definition

Function-definition-fix

Fixing function for function-definition structures.

Signature

(function-definition-fix x) → new-x

Arguments

x — Guard (function-definitionp x).

Returns

new-x — Type (function-definitionp new-x).

Definitions and Theorems

Function: function-definition-fix$inline

(defun function-definition-fix$inline (x)
 (declare (xargs :guard (function-definitionp x)))
 (let ((__function__ 'function-definition-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (b*
     ((header (function-header-fix (cdr (std::da-nth 0 x))))
      (precondition (maybe-expression-fix (cdr (std::da-nth 1 x))))
      (postcondition (maybe-expression-fix (cdr (std::da-nth 2 x))))
      (definer (function-definer-fix (cdr (std::da-nth 3 x)))))
     (list (cons 'header header)
           (cons 'precondition precondition)
           (cons 'postcondition postcondition)
           (cons 'definer definer)))
    :exec x)))

Theorem: function-definitionp-of-function-definition-fix

(defthm function-definitionp-of-function-definition-fix
  (b* ((new-x (function-definition-fix$inline x)))
    (function-definitionp new-x))
  :rule-classes :rewrite)

Theorem: function-definition-fix-when-function-definitionp

(defthm function-definition-fix-when-function-definitionp
  (implies (function-definitionp x)
           (equal (function-definition-fix x) x)))

Function: function-definition-equiv$inline

(defun function-definition-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (function-definitionp acl2::x)
                              (function-definitionp acl2::y))))
  (equal (function-definition-fix acl2::x)
         (function-definition-fix acl2::y)))

Theorem: function-definition-equiv-is-an-equivalence

(defthm function-definition-equiv-is-an-equivalence
  (and (booleanp (function-definition-equiv x y))
       (function-definition-equiv x x)
       (implies (function-definition-equiv x y)
                (function-definition-equiv y x))
       (implies (and (function-definition-equiv x y)
                     (function-definition-equiv y z))
                (function-definition-equiv x z)))
  :rule-classes (:equivalence))

Theorem: function-definition-equiv-implies-equal-function-definition-fix-1

(defthm
  function-definition-equiv-implies-equal-function-definition-fix-1
  (implies (function-definition-equiv acl2::x x-equiv)
           (equal (function-definition-fix acl2::x)
                  (function-definition-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: function-definition-fix-under-function-definition-equiv

(defthm function-definition-fix-under-function-definition-equiv
  (function-definition-equiv (function-definition-fix acl2::x)
                             acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-function-definition-fix-1-forward-to-function-definition-equiv

(defthm
 equal-of-function-definition-fix-1-forward-to-function-definition-equiv
 (implies (equal (function-definition-fix acl2::x)
                 acl2::y)
          (function-definition-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-function-definition-fix-2-forward-to-function-definition-equiv

(defthm
 equal-of-function-definition-fix-2-forward-to-function-definition-equiv
 (implies (equal acl2::x
                 (function-definition-fix acl2::y))
          (function-definition-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: function-definition-equiv-of-function-definition-fix-1-forward

(defthm
     function-definition-equiv-of-function-definition-fix-1-forward
  (implies
       (function-definition-equiv (function-definition-fix acl2::x)
                                  acl2::y)
       (function-definition-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: function-definition-equiv-of-function-definition-fix-2-forward

(defthm
     function-definition-equiv-of-function-definition-fix-2-forward
  (implies
       (function-definition-equiv acl2::x
                                  (function-definition-fix acl2::y))
       (function-definition-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)